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The cylinder problem:share your thoughts, ideas, questions,
and experiences; read what others have to say.

...we learned a lot from each other that we wouldn't have learned otherwise. I had worked the constant area problem abstractly, but I had never pictured it as taking a sheet of paper and cutting it up and pasting it together again, as the middle school teachers saw it.
-- Cynthia Lanius, Project Teacher

We chose this Experiment
with Volume because it involves a number of mathematical concepts and
problem-solving approaches that can lead to rich discourse in the classroom.

The Problem:

Form two cylinders from a rectangular piece of paper, one by
joining the long sides, one by joining the short sides. Which of these
cylinders will have greater volume, or will they hold the same amount?

In our experience, students are engaged by the mathematical demonstration
afforded by this investigation, the results of which seem counterintuitive
to many of them. The investigation also generates "aha" moments that motivate
students to focus their thinking.

When exploring this problem, students describe patterns, develop algorithms,
rules, and algebraic expressions, and explain their conclusions. Thus, the
lesson involves multiple stages of investigation: prediction, testing,
rejection or extension of hypotheses, discovering and exploring the underlying
mathematics, and making generalizations and proving results.

This kind of project offers many opportunities for fruitful conversation in the classroom, such as when students interpret a graph in Judith Koenig's class
[view clip], or when they provide
the reasoning for predictions in Susan Stein's class [view clip].

The students are actively engaged during these moments, negotiating and communicating
ideas with the teacher and with other students. The lesson incorporates manipulatives,
serving visual and tactile learners. In addition, because of the data that can be
generated, the problem is suitable for mathematical modeling. To help with the creation
of such models, we have the opportunity to use such technologies as graphing calculators
and spreadsheets, allowing students to recognize patterns without getting bogged down
in number-crunching.

The lesson also offers multiple entry points for a variety of levels, from
elementary school through calculus. Elementary students can explore the
family of cylinders that can be created from a single sheet of paper. Middle-level
students can use spreadsheets, calculate volumes, and relate them to the
physical models. Students of algebra and geometry can approach mathematical modeling
using linear, quadratic, cubic, and rational functions. Calculus students can
use derivatives with their mathematical models.

Finally, the problem presented in this lesson also has real life applications,
as Jon Basden demonstrates.